Modeling Improper Fractions

Why do students who may have succeeded in mathematics in the lower grades struggle when they get into third grade or above?  Students may do quite well with whole number mathematics, but once we start working with fractions, often tend to struggle. This struggle, I believe is due to the lack of understanding of fractions we teachers have ourselves.  Most textbooks never go deeper than the understanding that a fraction as a part of a whole.
Therefore, we often only teach to this level of our own understanding.  Fraction introduction at earlier grades and level of understanding of fractions are heavy in the Common Core Standards for mathematics.  There is evidence to support that the reason our students do not do well in algebra stems back to this surface level understanding of fractions our students learn in the elementary school.  Therefore, it's time for us all to increase our level of understanding of fractions.  I always say,

"The fact that we don't understand fractions better is not our fault, it's how we were taught, but... now that we know we don't have a deep understanding, it is negligent if we don't choose to study and learn so we can deepen our students mathematical understanding."

Often times when students encounter improper fractions, such as 5/2 we teach them to divide five by two, which gives them the answer 2 and 1/2.  My experiences lead me to believe although students can do the division, they have no ideas why the answer is 2 and ½  or if they divided wrong and get 20 ½ , they wouldn't even know if their answer is incorrect.  So why is 5/2 the same as 2 ½ .  Let's look first at the division model we are more familiar with:

 

Before we just ask students to divide the numbers, they must first understand that the line under the five and above the two means to divide.  They must also first start by using concrete models, moving to a pictorial representation like the one above, and then move to the division operation.

Now, let's look at another model of 5/2 that we may not be as familiar with.  This model is repeated addition or iteration.  Research by Post, Washsmuth, Legh, & Behr, 1985 and Siebert and Gaskin in 2006, provides evidence that an iterative notion of fractions, one that views a fraction such as ¾ as a count of three parts called fourths, is an important idea for children to develop.  If we examine that model with our improper fraction 5/2 then 5/2 is the same as or equal to ½  + ½  + ½  + ½  + ½. How can we model this?  A number line is a very useful tool.  This shows why ½ added five times gets you to 2 ½ and why ½ + ½ = 1 whole plus another ½ + ½ gets you another whole, giving you two wholes, then there is another ½ left over to get the answer 2 ½.  

 

Iterating an also be shown with other length models as well as area models. Start the iterating model with fractions less than one is the precursor for using it with improper fractions.

As a teacher, we need to add these two models of understanding for our students.  Both represent a unique model for an improper fraction.  It is also important for us to understand that we must take the time to spend more time on the conceptual understanding of fractions before we ask students to use calculations that make little to no sense to them.  If you have other ideas for fractions, I would love to hear more ideas at michelleflaming143@gmail.com